Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties

We study modular differential equations (MDEs) of high orders for weak Jacobi forms and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for the elliptic genus of six-dimensional manifolds with a trivial first Chern class. We prove that the minimal possible order of the MDE for the elliptic genus of a strict six-dimensional Calabi--Yau variety is four, and find MDEs of order 7 for hyperkähler varieties of dimension 6. The latter MDEs correspond to the generic case. The non-generic weak Jacobi forms of weight 0 and index 3 form a divisor that contains two cubic plane curves in the coefficient space.